Asymptotic performance of projection estimators in standard and hyperbolic wavelet bases

Florent Autin, Gerda Claeskens and Jean-Marc Freyermuth

No 504649, Working Papers of Department of Decision Sciences and Information Management, Leuven from KU Leuven, Faculty of Economics and Business (FEB), Department of Decision Sciences and Information Management, Leuven

Abstract: We provide a novel treatment of the ability of the standard (wavelet-tensor) and of the hyperbolic (tensor product) wavelet bases to build nonparametric estimators of multivariate functions. First, we give new results about the limitations of wavelet estimators based on the standard wavelet basis regarding their inability to optimally reconstruct functions with anisotropic smoothness. Next, we provide optimal or near optimal rates at which both linear and non-linear hyperbolic wavelet estimators are well-suited to reconstruct functions from anisotropic Besov spaces and subsequently we characterize the set of all the functions that are well reconstructed by these methods with respect to these rates. As a first main result, we furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, in particular by showing a stronger exposure of linear methods to the curse of dimensionality. Second, we propose an adaptation of the well known block thresholding method to a hyperbolic wavelet basis and show its ability to estimate functions with anisotropic smoothness at the optimal minimax rate. Therefore, we prove the pertinence of horizontal information pooling even in high dimensional settings. Numerical experiments illustrate the finite samples properties of the studied estimators.

Keywords: Anisotropy; Besov spaces; Information pooling; Linear and non linear methods; Multivariate wavelet basis; Thresholding (search for similar items in EconPapers)
Date: 2015
References: Add references at CitEc
Citations:

Published in FEB Research Report KBI_1518

Downloads: (external link)
https://lirias.kuleuven.be/retrieve/330106 Asymptotic performance of projection estimators in standard and hyperbolic wavelet bases (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:ete:kbiper:504649

Access Statistics for this paper

More papers in Working Papers of Department of Decision Sciences and Information Management, Leuven from KU Leuven, Faculty of Economics and Business (FEB), Department of Decision Sciences and Information Management, Leuven
Bibliographic data for series maintained by library EBIB ().